Chapter 2, Problem 2.14P

(a)

To determine

The proof of $E\left[g\left(x\right)\right]\le g\left[E\left(x\right)\right]$ (Jensen’s Inequality)

(b)

To determine

The proof of $E\left[g\left(x\right)\right]\ge g\left[E\left(x\right)\right]$

(c)

To determine

The proof of $E\left(x\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}\left[1-F\left(x\right)\right]dx}$

(d)

To determine

The proof of $P\left(x\ge t\right)\le \frac{E\left(x\right)}{t}$ (Markov’s Inequality)

(e)

To determine

1. The proof that $f\left(x\right)=2x^{-3}$ for $x\ge 1$ is a proper PDF
2. $F\left(x\right)$ for this PDF
3. $E\left(x\right)$ for this PDF using the result of part (c)
4. The proof that Markov’s inequality holds for this function

(f)

To determine

1. The proof that $f\left(x\right)=\frac{x^2}{3}$ for $-1\le x\le 2$ is a proper PDF
2. The value of $E\left(x\right)$
3. The probability that $-1\le x\le 0$
4. The value of $f\left(x|A\right)$ , where $A$ is the event $0\le x\le 2$
5. The value of $E\left(x|A\right)$
6. Intuitive explanation of the results